2. Find a power series representation for the following function and determine the interval of convergence. 2-1 +2 f(x)= 0 Answer: 1- Σ n=0 (-1)"3" 2n+1 ; I = (-2,2) 3. Find a power series representation for the following function and determine the radius of convergence. f(x) = ln(4-x) Answer: In(4) - ; R=4 n4 4. Find the Maclaurin series for the following function. Also state the associated radius of convergence. f(x)= 1 (1-x)2 00 Answer: (n+1)x"; R=1 n=0 5. Find the Taylor series for the following function centered at the given value of a. f(x) = Answer: a = -3 - 0 n=0 (+3) 3+1 ; R = 3 6. Find the Taylor series for the following function centered at the given value of a. f(x) = cos(2), a= " 2 Answer: Σ (-1)+1 (2n+1)! 2n+1 ; R=00 n=0 7. Find the Maclaurin series for the following function. f(x)= 1- cos(22) 2 (HINT: : cos(x)=(-1)) (2n)! n=0 00 Answer: (-1)+12-1 (2n)! 8. Evaluate the following indefinite integral as an infinite series. 'tan (r) dr HIN 0 tan(x)=(-1)+1 2n+1 п 00 Answer: C+ n=0 (-1)"2+5 (2n+1)(2n+5) 9. Use the binomial series to expand the following function as a power series. Also state the radius of convergence. 1 f(x)= (2+2)3 (HINT: (1+r)*=) = (1+2)=(")=" "where (h) = k(k-1). (kn+1)) ...(k - по Answer: (-1)(n+1)(n+2)z" n=0 2n+4 ; R=2 10. Use the binomial series to expand the following function as a power series. Also state the radius of convergence. f(x)=√4(HINT: (1+2)* - Σ (^) z Answer: + IM8 = n=0 " where (*) = (k-1) (n+1) n! ( (-1) 1-3.5(2n-1)+1 23n+1n! ; R=2 11. Sketch the curve of the given parametric equations to plot points using the grid be- low. Indicate with an arrow the direction in which the curve is traced as t increases: z=-3, y=t+2, -3≤1≤3. Answer: t=1, (-2, 3) 4 1=3, (6,5) t=-1, 2+ (-2,1) 2 -4-20 -2+ x 1=-3, (6,-1) 12. Based on the problem 11, eliminate the parameter to find a Cartesian equation of the curve. Answer: =y2-4y+1, -1≤ y ≤5 13. Sketch the curve of the given parametric equations to plot points using the grid on the next page. Indicate with an arrow the direction in which the curve is traced as t increases: z=sin(t), y=1-cos(t), 0 ≤t≤2. Answer: On next page 14. Based on the problem 13, eliminate the parameter to find a Cartesian equation of the curve. Answer: r²+(y-1)²=1 15. Find an equation of the tangent to the following curve at the point corresponding to the given value of the parameter. =1+ √t, y=e, (2, e) Answer: y=4ex - 7e 16. Find an equation of the tangent to the following curve at the point corresponding to the given value of the parameter. z=e' sin(t), y=e, t=0 2 Answer: y=x+1 17. Find a polar equation for the curve represented by the following Cartesian equation. 2= = 4y2 Answer r= =cot (0) csc(0) 18. Find the slope of the tangent line to the following polar curve at the point specified by the value of 0. T = COS 3 י 0=" Answer: -√√3 19. Find the area of the region that lies inside the following first curve and outside the second curve. r 4 cos(0), r = 2 = 4 Answer: +2√3 3 20. Find the area of the region that lies inside both of the following curves. r=3+2 cos(0), r=3+2sin(0) Answer: 11-12√√2